*> \brief <b> SSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
* M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ, RANGE
* INTEGER IL, INFO, IU, LDZ, M, N
* REAL ABSTOL, VL, VU
* ..
* .. Array Arguments ..
* INTEGER IFAIL( * ), IWORK( * )
* REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SSTEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a real symmetric tridiagonal matrix A. Eigenvalues and
*> eigenvectors can be selected by specifying either a range of values
*> or a range of indices for the desired eigenvalues.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBZ
*> \verbatim
*> JOBZ is CHARACTER*1
*> = 'N': Compute eigenvalues only;
*> = 'V': Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all eigenvalues will be found.
*> = 'V': all eigenvalues in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix. N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is REAL array, dimension (N)
*> On entry, the n diagonal elements of the tridiagonal matrix
*> A.
*> On exit, D may be multiplied by a constant factor chosen
*> to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is REAL array, dimension (max(1,N-1))
*> On entry, the (n-1) subdiagonal elements of the tridiagonal
*> matrix A in elements 1 to N-1 of E.
*> On exit, E may be multiplied by a constant factor chosen
*> to avoid over/underflow in computing the eigenvalues.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is REAL
*> If RANGE='V', the lower bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is REAL
*> If RANGE='V', the upper bound of the interval to
*> be searched for eigenvalues. VL < VU.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest eigenvalue to be returned.
*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*> ABSTOL is REAL
*> The absolute error tolerance for the eigenvalues.
*> An approximate eigenvalue is accepted as converged
*> when it is determined to lie in an interval [a,b]
*> of width less than or equal to
*>
*> ABSTOL + EPS * max( |a|,|b| ) ,
*>
*> where EPS is the machine precision. If ABSTOL is less
*> than or equal to zero, then EPS*|T| will be used in
*> its place, where |T| is the 1-norm of the tridiagonal
*> matrix.
*>
*> Eigenvalues will be computed most accurately when ABSTOL is
*> set to twice the underflow threshold 2*SLAMCH('S'), not zero.
*> If this routine returns with INFO>0, indicating that some
*> eigenvectors did not converge, try setting ABSTOL to
*> 2*SLAMCH('S').
*>
*> See "Computing Small Singular Values of Bidiagonal Matrices
*> with Guaranteed High Relative Accuracy," by Demmel and
*> Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*> M is INTEGER
*> The total number of eigenvalues found. 0 <= M <= N.
*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is REAL array, dimension (N)
*> The first M elements contain the selected eigenvalues in
*> ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is REAL array, dimension (LDZ, max(1,M) )
*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*> contain the orthonormal eigenvectors of the matrix A
*> corresponding to the selected eigenvalues, with the i-th
*> column of Z holding the eigenvector associated with W(i).
*> If an eigenvector fails to converge (INFO > 0), then that
*> column of Z contains the latest approximation to the
*> eigenvector, and the index of the eigenvector is returned
*> in IFAIL. If JOBZ = 'N', then Z is not referenced.
*> Note: the user must ensure that at least max(1,M) columns are
*> supplied in the array Z; if RANGE = 'V', the exact value of M
*> is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1, and if
*> JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*> IFAIL is INTEGER array, dimension (N)
*> If JOBZ = 'V', then if INFO = 0, the first M elements of
*> IFAIL are zero. If INFO > 0, then IFAIL contains the
*> indices of the eigenvectors that failed to converge.
*> If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, then i eigenvectors failed to converge.
*> Their indices are stored in array IFAIL.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup realOTHEReigen
*
* =====================================================================
SUBROUTINE SSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
$ M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE
INTEGER IL, INFO, IU, LDZ, M, N
REAL ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER IFAIL( * ), IWORK( * )
REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
CHARACTER ORDER
INTEGER I, IMAX, INDIBL, INDISP, INDIWO, INDWRK,
$ ISCALE, ITMP1, J, JJ, NSPLIT
REAL BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
$ TMP1, TNRM, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANST
EXTERNAL LSAME, SLAMCH, SLANST
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SSCAL, SSTEBZ, SSTEIN, SSTEQR, SSTERF,
$ SSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -7
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -9
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
$ INFO = -14
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSTEVX', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 )
$ RETURN
*
IF( N.EQ.1 ) THEN
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = D( 1 )
ELSE
IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
M = 1
W( 1 ) = D( 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = SLAMCH( 'Safe minimum' )
EPS = SLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
IF ( VALEIG ) THEN
VLL = VL
VUU = VU
ELSE
VLL = ZERO
VUU = ZERO
ENDIF
TNRM = SLANST( 'M', N, D, E )
IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / TNRM
ELSE IF( TNRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / TNRM
END IF
IF( ISCALE.EQ.1 ) THEN
CALL SSCAL( N, SIGMA, D, 1 )
CALL SSCAL( N-1, SIGMA, E( 1 ), 1 )
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
*
* If all eigenvalues are desired and ABSTOL is less than zero, then
* call SSTERF or SSTEQR. If this fails for some eigenvalue, then
* try SSTEBZ.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
CALL SCOPY( N, D, 1, W, 1 )
CALL SCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
INDWRK = N + 1
IF( .NOT.WANTZ ) THEN
CALL SSTERF( N, W, WORK, INFO )
ELSE
CALL SSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ), INFO )
IF( INFO.EQ.0 ) THEN
DO 10 I = 1, N
IFAIL( I ) = 0
10 CONTINUE
END IF
END IF
IF( INFO.EQ.0 ) THEN
M = N
GO TO 20
END IF
INFO = 0
END IF
*
* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
INDWRK = 1
INDIBL = 1
INDISP = INDIBL + N
INDIWO = INDISP + N
CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
$ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ),
$ WORK( INDWRK ), IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL SSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
$ Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL,
$ INFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
20 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors.
*
IF( WANTZ ) THEN
DO 40 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 30 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
30 CONTINUE
*
IF( I.NE.0 ) THEN
ITMP1 = IWORK( INDIBL+I-1 )
W( I ) = W( J )
IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
W( J ) = TMP1
IWORK( INDIBL+J-1 ) = ITMP1
CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
IF( INFO.NE.0 ) THEN
ITMP1 = IFAIL( I )
IFAIL( I ) = IFAIL( J )
IFAIL( J ) = ITMP1
END IF
END IF
40 CONTINUE
END IF
*
RETURN
*
* End of SSTEVX
*
END